1

An Ankle-Foot Prosthesis EmulatorCapable of Modulating Center of Pressure

Vincent L. Chiu, Alexandra S. Voloshina, Steven H. Collins*

Abstract—Objective: Several powered ankle-foot prostheseshave demonstrated moderate reductions in energy expenditureby restoring pushoff work in late stance or by assisting withbalance. However, it is possible that heel collision work or centerof pressure trajectory modulation could provide even furtherimprovements in user performance. Here we describe the designof a prosthesis emulator with two torque-controlled forefootdigits and a torque-controlled heel digit. Independent actuationof these three digits can modulate the origin and magnitude ofthe total ground reaction force vector. Methods: The emulatorwas designed to be compact and lightweight while exceeding therange of motion and torque requirements of the biological ankleduring walking. We ran a series of tests to determine torquemeasurement accuracy, closed-loop torque control bandwidth,torque tracking error, and center of pressure control accuracy.Results: Each of the three digits demonstrated less than 2 Nmof RMS torque measurement error, a 90% rise time of 19 ms,and a bandwidth of 33 Hz. The untethered end-effector has amass of 1.2 kg. During walking trials, the emulator demonstratedless than 2 Nm of RMS torque tracking error and was ableto maintain full digit ground contact for 56% of stance. Infixed, standing, and walking conditions, the emulator was able tocontrol center of pressure along a prescribed pattern with RMSerrors of about 10% of the length of the pattern. Conclusion:The proposed emulator system met all design criteria and caneffectively modulate center of pressure and ground reaction forcemagnitude. Significance: This emulator system will enable rapiddevelopment of controllers designed to enhance user balance andimprove user energy expenditure. Experiments conducted usingthis emulator will identify beneficial control behaviors that can beimplemented on autonomous devices, in turn improving mobilityand quality of life of individuals with amputation.

Index Terms—Biomechanics, ankle-foot prosthesis, center ofpressure.

I. INTRODUCTION

There were about 1 million individuals in the United Statesliving with a lower limb amputation in 2005, with that numberprojected to more than double by 2050 [1]. These individualssuffer from asymmetries and reduced locomotor performance.Specifically, those with unilateral transtibial amputation walkwith asymmetrical step lengths to compensate for lack ofpushoff [2] and walk 11% slower compared to individualswithout amputation [3]. Over uneven terrain, such as tall grassor a loose rock surface, individuals with transtibial amputationtake shorter, wider steps and consume 17% more energycompared to individuals without amputation [4], [5]. These

This work was supported by the National Science Foundation, Alexandria,VA, USA, under Grant No. CBET-1511177. Asterisk indicates correspondingauthor.

V. L. Chiu, A. S. Voloshina, and *S. H. Collins are with the Department ofMechanical Engineering, Carnegie Mellon University, Pittsburgh PA 15213USA and the Department of Mechanical Engineering, Stanford University,Stanford, CA 95014 USA (e-mail: stevecollins@stanford.edu)

penalties may be due to reduced center-of-pressure control[6] and greater instability during walking with perturbationsand over uneven terrain [7], [8]. Individuals with lower limbamputation are also at a higher risk of falling, with about50% of the population falling each year [9]. These falls occurin both the elderly as well as young, active individuals withamputation and often result in complications such as infection[10]. As a result, about 50% of individuals with lower limbamputation report fear of falling, and suffer from reducedsocial activity and a decrease in quality of life [9].

Several active ankle-foot prostheses have been developedwith the goal of improving locomotor performance. Suchdevices, whether for commercial or research purposes, havelargely focused on pushoff work in late stance, as it hasbeen suggested that reduced pushoff from the affected limbmay increase step-to-step energy collision losses and, in turn,metabolic rate [3], [11]. As such, several devices have beendesigned to restore powered pushoff to the user [12], [13],[14], with one such device showing an 11% reduction inmetabolic rate using active plantarflexion control as dictatedby a neuromuscular model [15]. It is possible that providingonly powered pushoff assistance is not effective in reduc-ing energy expenditure in individuals with amputation [16],as finer details such as pushoff timing may also influenceperformance measures [17]. As a result, some methods haveexplored the effects of balance assistance on performance [18],[19]. Hand-tuned assistance of step-by-step plantarflexion andinversion/eversion control in response to center of mass motionhas even demonstrated 9% reductions in energy expenditure inindividuals with amputation [20]. However, our understandingof the effects of balance assistance on energy expenditureand overall locomotor performance is still limited. In order tobetter understand how to provide effective balance assistance,we must develop devices and prosthesis behaviors capableof controlling balance-related measures such as center ofpressure.

New prosthesis behaviors can be quickly developed usingprosthesis emulator systems. Inspired by earlier experimentaltethered devices (such as [21], [22], [23], [24]), these lab-basedsystems consist of a generalized prosthetic device tethered tooff-board motors and computers dictating device control inreal-time [25]. Compared to commercial devices, prosthesisemulators allow new behaviors to be tested without designingspecialized hardware for each new behavior. Instead, newbehaviors can be written and implemented on the emulatorsystem, allowing for rapid iteration and faster discovery ofhelpful behaviors. We have worked extensively with tetheredemulator systems, using them to characterize and optimizeassistive ankle torque profiles for both exoskeletons and pros-

2

theses (e.g. [26], [27], [28], [29]). Using a one degree offreedom prosthetic emulator, we were able to show that devicework rate does not necessarily affect energy expenditure inindividuals with unilateral transtibial amputation [16], [30].However, providing assistance in the medio-lateral directionusing a two degree of freedom emulator did lead to a reductionin metabolic cost during walking [20]. Since inversion/eversioncontrol exerts some control over the foot center of pressuretrajectory, this suggests that a more direct center of pressurecontrol approach could lead to even larger improvements inuser balance and energy expenditure.

Here we outline the design of a prosthesis emulator intendedto assist in the development of prosthesis behaviors focusedon improving balance. It is equipped with three degrees offreedom which allow us to modulate the center of pressureand ground reaction force magnitude.We performed benchtoptesting to determine strain gauge measurement accuracy andclosed-loop torque control bandwidth. We also tested theaccuracy of the center of pressure estimation of the emulator,as well as torque tracking accuracy during normal walking ofa person with a unilateral transtibial amputation. We expectthis emulator system to enable experiments that will evaluateprosthesis behavior on human locomotion and inform futureprosthesis design.

II. MECHANICAL DESIGN

We designed a prosthesis emulator system with three sep-arately controlled digits, which allow us to modulate theground reaction force origin location (center of pressure) andmagnitude. Each digit generates a ground reaction force, withthe total ground reaction force vector calculated as the vectorsum of the three vectors. Modulating the center of pressure andground reaction force magnitude is subject to some limitations,addressed in Section V. The whole system comprises a three-motor actuation unit, a computer, and an end-effector worn bythe user (Fig. 1).

The primary components of the end-effector include twoforefoot digits, a heel digit, two frame halves, and a bridge(Fig. 2). Each half of the frame contains a revolute joint fora forefoot digit, with the heel digit located between the twoframe halves. A bridge connects the two frame halves andprovides a mounting location for a standard pyramid adapter.Each digit acts as a lever: when one end is pulled up by theBowden cable, the other end exerts a force on the ground(Fig. 3). The digits are sized such that the ground contactlocations of the heel and forefoot digits relative to the effectiveankle joint correspond to anthropomorphic locations of theheel and metatarsals.

A. Geometry

The geometry of the end-effector was designed to generatedesired forces and torques within a desired range of motionwhile adhering to the constraints of the off-board motor unit.The forces and torques were selected to exceed the require-ments of a user with weight of 110 kg walking at 1.25 m/swhile the size was designed to match that of an average malefoot [31].

ProsthesisEnd-Effector

Bowden CablesMotors

Computer

Pyramid Adapter

Bowden Cable(Inner Rope)

Bowden Cable(Outer Conduit)

HeelDigit

ForefootDigit

DataCable

(a)

(b)

Fig. 1. The emulator system consists of powerful off-board motors, flexibleBowden cable and data tethers, an end-effector worn by the user, and acomputer (a). Force is transferred from the Bowden cables to each of thethree digits: a heel and two forefoot digits. The end-effector connects to theuser via a standard pyramid adapter (b).

1

3 4

2

Fig. 2. The six primary components of the end effector include two framehalves (light grey), a connecting bridge (dark grey), two forefoot digits (greenand orange), and a heel digit (blue).

3

(a) (c)(b)

Fig. 3. Bowden cables provide unidirectional actuation for two forefoot digits(a),(b), and a heel digit (c). Counter-actuation is provided by retraction springs(Fig. 6).

We relied on kinetic and kinematic data of normal walking[32] to guide the geometry of the end-effector. The geometrywas parameterized into: lengths of the lever arms, anglesbetween the digit and lever arms, and the location of the heeljoint relative to the forefoot joint. We observed the effect ofvarying these parameters on the Bowden cable forces requiredto generate the desired motion, as well as whether the leverarms collided with the ground. The forefoot digits spannedfrom the metatarsal joints to the ankle, which was used as thelocation of the forefoot joint. The heel digit spanned from thecalcaneus to the variable heel joint. The resulting geometryyielded a vertical range of translation of 0.050 m, maximumdorsiflexion of 32◦, maximum plantarflexion of 26◦, andmaximum inversion/eversion of ±57◦. These measurementscan be visualized in Fig. 4 as the angle between the groundand the plane normal to the prosthesis pylon.

The end-effector is designed to be lightweight while stillbeing able to generate the desired forces and torques. Toaccomplish this goal, the major components of the end-effectorwere CNC milled from 7075 aluminum alloy, which offershigh specific strength and is easy to machine. Since the frameand digits are primarily loaded in bending, we incorporatedpocket cutouts and tapered cross-sectional heights into thedesign of these parts. Pocketing and tapering remove materialwhere bending stress is low and reduce the mass of the end-effector while maintaining strength. The resulting mass of theend-effector is 1.2 kg, not including the tether, and is designedfor individual forefoot torques up to 70 Nm and heel torquesup to 100 Nm.

B. Sensors and Control

Three rotary encoders and three strain gauges measure thejoint angles and torques of the end-effector (Fig. 5). Eachdigit features a strain gauge (Omega, Stamford, CT, USA)in a full-bridge configuration, utilizing dual grids in parallelon the top and bottom surfaces. This configuration is highlysensitive to longitudinal bending strain while rejecting axialstrain and temperature effects.

A rotary encoder is mounted coaxially with each digit jointto measure joint angles. To avoid damaging the encoders byexternal impacts, we designed the encoders to fit on the insidesurfaces of the frame. We selected the RLS RM08 (RLS,Komenda, Slovenia) for its small footprint and extremely lowprofile height.

The sensor wires are routed to a connection panel featuringDB9 and DB15 terminal blocks. This panel offers strain relief

(a) (b) (c)

Fig. 4. Rotating both forefoot digits in one direction and the heel digit in theopposite direction moves the emulator vertically (a) Rotating all three digitsin the same direction rotates the end-effector in the sagittal plane (b). Rotatingthe forefoot digits in opposite directions rotates the emulator in the frontalplane (c).

Fig. 5. Each digit is equipped with a rotary encoder and strain gauges onthe top and bottom surfaces. Strain gauges are rendered in blue and encoderswith their corresponding shafts are rendered in orange. Sensor wires routeto a connection panel utilizing DB9 and DB15 terminal blocks, rendered ingreen.

and pull protection for the sensors and allows standard D-Subcables to be used for data transmission. These cables are easilyreplaced in the event of a cable failure.

Our computer, or real-time target machine (Speedgoat,Liebefeld, Switzerland), samples the analog signals of theencoders and strain gauges at 1000 Hz. We then filter thesignals down to 100 Hz using a 2nd order Butterworth filter.To further improve the signal-to-noise ratio of our analogsignals, we use shielded D-Sub cables to reduce the effectsof electromagnetic interference. We also found it helpful toelectrically isolate the motor unit from the end-effector byplacing an insulating washer between the Bowden cable outerconduit and the end-effector frame (Fig. 6a).

Emulator system behavior is governed via a two-tier controlarchitecture comprising a high-level controller and a low-level controller. The high-level controller defines the overallbehavior of the emulator and determines desired digit torques.It operates on a state machine that detects gait events such

4

as heel contact, forefoot contact, heel liftoff, and toe-off. Byvarying torque patterns based on changing gait events, thehigh-level controller can be programmed to emulate a varietyof behaviors, including impedance controllers [33], neuromus-cular controllers [34], quasi-active controllers [35], energyrecycling behaviors [36], or balance assistance controllers [18].

The low-level controller accepts desired torques from thehigh-level controller and commands motor velocity to trackthe desired torque. The Bowden cable acts as a series elas-tic element, largely dictating the relationship between motordisplacement and end-effector torques. The controller utilizesa proportional error term Kp, a motor velocity damping termKd , and a feedforward iterative learning term Kl :

θ̇motor = Kp(τdes − τmeas)−Kd θ̇motor +Kl θ̇learn

where τdes and τmeas are desired and measured digit torquesand θ̇motor is the angular velocity of the motor attached tothat digit. The proportional error term allows the controllerto track desired torque trajectories while the damping termreduces overshoot in the torque tracking response. In order toreduce the tracking error that arises from nonlinearities in thesystem, we also implemented a feedforward iterative learningterm. Using this learning term in conjunction with PD controlhas been shown to reduce root-mean-square (RMS) torquetracking errors in tethered exoskeleton emulators up to 84%compared to PD control alone [37]. To do so, the learningterm first parameterizes the walking stride in time startingfrom a gait event such as initial heel contact. The controllerthen accumulates the error between desired and measuredtorques for a particular time step across all previous strides.This accumulated error is then converted to feedfoward motorvelocity compensation by multiplying the error by a gain. Overtens of strides the learning term reduces torque tracking errorsthat occur at the same time step per stride.

C. Additional Features

We implemented several features to increase the lifespanof the end-effector. Each digit is equipped with a hard stopin either direction of rotation (Fig. 6a). These hard stops aredesigned to withstand the impact of the digit if the motorproduces larger torques than expected or if there is too muchslack in the inner rope and the digit bottoms out.

A pulley is located below each Bowden outer conduittermination (Fig. 6a). This pulley guides the inner rope toexit the outer conduit perpendicularly before routing towardsthe inner rope termination on the digit. A perpendicular exitprevents the inner rope from rubbing on the frame and extendsthe lifespan of the rope.

The end-effector frame has strain relief for each Bowdencable outer conduit (Fig. 6a). A 3D printed conduit terminationfeatures a flexible urethane insert that encourages a gentlecurve in the outer conduit, reducing the stress concentrationcompared to a sharp bend.

The ends of each digit feature 3D printed ground contactpads (Fig. 6b). These pads provide a larger mounting locationfor rubber, which gives the end-effector additional tractionduring walking. The rubber strips are glued onto the pads

ConnectionPanel

Bowden CableStrain Relief

RetractionSpring

Strain GaugeCover

GroundContact

Pad

PulleyHard Stop

(b)

Pulley

Hard Stop

UrethaneInsert

Elastic Insert

Retraction SpringHook

RubberTraction

Strip

ElectricallyInsulatingWasher

Strain Gauge PocketCutout

Strain GaugeCover Protrusion

(c)

(d)

(a)

Fig. 6. Features implemented in order to improve the lifespan of the end-effector (a). The Bowden cable strain relief reduces the bending fatigue of theBowden cable outer conduit (b). The ground contact pads feature a limitedamount of rotation to reduce the wear of the rubber traction strips (c). Thestrain gauge covers protect the strain gauges from impacts. The cover featuresprotrusions that fit inside pockets on the digit to stay in place (d).

and are replaced when worn. During walking, the heel andforefoot pads move relative to each other, which causes therubber to scrub along the ground, accelerating its wear. Tomitigate this issue, we designed the pads to have a limitedamount of rotation in the sagittal plane on its mounting pointon the digit. Elastic inserts control the amount of rotation andreturn the pad to a neutral angle when the digit is off theground.

Each strain gauge is protected by a 3D printed cover toprevent impacts from damaging the sensor. The cover usesa clamshell design and features protrusions to hold itself in

5

0 1 2

Mag

nitu

de (d

B)

Frequency (Hz)

Phas

e (d

eg)

Applied Torque (Nm)

Mea

sure

d To

rque

(Nm

)

Time (ms)

Torq

ue (N

m)

-4-3-2-10

10 10 10-200-150-100

-500

0

20406080

100120

0 5 10 15 20 25

10

30

50

70

20 40 60 80 100 120

-3 dB crossing:38 Hz

30° phase margin:33 Hz

RMSE: 1.7 Nm90% crossing:

19 ms

(a) (b)

(c)

Fig. 7. Benchtop testing demonstrated low measurement error, low risetimes, and high torque control bandwidth. All three digits reported nearlyidentical results. Measurement accuracy was determined by comparing knownapplied torques to measured torques. Measurement error is less than 3% ofpeak torques (a). Step response testing was performed by instantaneouslychanging the desired torque and measuring how long it took the system torespond. On average it took 19 ms to reach 90% of the desired torque (b).Bandwidth testing was performed by commanding a series of 40 sine waves.The calculated bandwidth is 33 Hz, limited by a 30◦ phase margin (c).

place using the walls of the pockets on the digits (Fig. 6c).Since the Bowden cable only provides unidirectional ac-

tuation, rubber bands are used as retraction springs to takeup slack in the inner rope (Fig. 6). These rubber bandsspan between hooks located on the ground contact pads andmounting points on the frame.

III. BENCHTOP TESTING

We experimentally validated force and torque capabilitiesof the emulator system through a series of benchtop tests.We verified torque measurement accuracy, closed-loop torquestep response times, and closed-loop torque control bandwidth.During all tests we fixed the end-effector in a rigid frame thatprevented movement of the digits.

To determine torque measurement accuracy, we first cali-brated the strain gauges to determine a relationship betweenapplied torque and output voltage. A validation trial was thenperformed by comparing known applied torques to measuredtorques. The strain gauges were calibrated one at a time byapplying known forces to the end of each digit and recordingthe resulting strain gauge voltages. The voltages were recordedacross six force conditions, including a zero load condition,in both loading and unloading directions to account for anyhysteresis. A linear relationship was then determined betweenvoltage and applied torque. The validation trial revealed torquemeasurement error with a root-mean-square value of 1.7 Nm,or less than 3 % of the expected peak torque (Fig. 7a).

To determine closed-loop step response, we used a PDcontroller to control digit torques and kept controller gainsconstant across all testing. In the step response test, desiredtorque was instantaneously changed from 5 Nm to 70 Nm. Thetest began at 5 Nm in order to remove slack in the Bowdencables. We conducted 10 trials per digit and averaged 90% risetimes over all trials to determine the step response for eachdigit. The step response time was found to be 19± 0.56 msfor all three digits (Fig. 7b).

To test closed-loop torque control bandwidth, digit torquestracked sine waves with frequencies ranging from 1 to 40 Hzin increments of 1 Hz. The sine waves were centered around35 Nm with an amplitude of 15 Nm. The controller trackedeach sine wave for 3 s before the frequency was incremented.We averaged amplitude ratios and phase shift angles for eachfrequency over ten trials and then generated a Bode plot todetermine the frequency response of the system. The closed-loop torque control bandwidth was identical for all three digitsand was found to be 33± 0.31 Hz, limited by a 30◦ phasemargin (Fig. 7c).

Closed-loop torque control bandwidth was found to behigher than similar tethered [38] and untethered [39] devices.Fast step response times and high torque control bandwidthsallow for accurate torque control and high fidelity hapticemulation [40]. Specifically, this allows for a more realisticrendition of a behavior we would choose to emulate.

IV. WALKING TRIAL

To validate the torque tracking capabilities of the emulator,one participant (male, 74 kg, 60 years of age) with a unilateraltranstibial amputation walked while using the emulator system.The participant provided written informed consent prior toparticipating in the study and was fitted to the end-effectorby a licensed prosthetist. All experimental procedures wereapproved by the Stanford University Institutional ReviewBoard.

The end-effector was controlled such that each digit in-dependently emulated a virtual spring: τdes = k(θ − θneutral),where τdes is the desired torque for the digit, k is the stiffnessconstant adjusted to user preference, and θneutral is the restingangle of the digit. The resting angles for each digit are suchthat when no force is applied to the end-effector, the end-effector frame is parallel to the ground at a user-selectedheight.

The participant walked at 1 m/s on a level treadmill for5 min, recording 216 steps during the trial. The RMS torquetracking errors for the left forefoot, right forefoot, and heeldigits were 1.8 Nm, 1.3 Nm, and 1.3 Nm, respectively (Fig. 8).These values were about 2.5% of the peak torques recordedfor all three digits (71 Nm, 56 Nm, and 52 Nm) during thetrial. Due to the compensation provided by the feedforwarditerative learning term (described in Section II B), the RMSEvalues gradually reduced over the duration of the trial. TheRMSE values for the first 50 steps of the trial were 2.1, 1.5,and 1.4 Nm for the three digits but reduced to 1.6, 1.1, and1.1 Nm for the last 50 steps.

The RMSE values when the digits are being loaded (torqueis increasing) are higher than when the digits are being

6

Loading

Desired 1 Std Dev

Mean

-0.35 -0.3 -0.25 -0.2 -0.15Digit angle (rad)

0

10

20

30

40

50

60

70

80

Dig

it to

rque

(Nm

)

-0.35 -0.3 -0.25 -0.2 -0.15Digit angle (rad)

Unloading

(a) (b)

Fig. 8. Average torque tracking results over 216 steps from a 5 min walkingtrial with an individual with unilateral transtibial amputation. RMSE trackingerror was 1.8, 1.3, and 1.3 Nm for the two forefoot digits and the heel digit,respectively. Only left forefoot digit torque tracking graphs are shown, as theplots for the other two digits are visually very similar. The black line andshaded blue area indicate the mean and one standard deviation range of themeasured torque. The orange line indicates the desired torque, with the slopeof the line corresponding to the stiffness constant of the digit. Torque trackingdata is split into plots where the digit is being loaded i.e. torque is increasing(a) and where the digit is being unloaded i.e. torque is decreasing (b).

unloaded (torque is decreasing). This is primarily seen in themismatch between the desired and produced torque at thebeginning of the step (Fig. 8a). This error is due to the fact thatwhen a digit first contacts the ground, the angle of the digitchanges quickly, which results in a rapid change of desiredtorque. The low-level controller is unable to respond quicklyenough and temporarily lags behind. The feedforward iterativelearning term should be able to compensate for such errors,but due to variations in timing on each step, the learning termis unable to accurately predict the spike in desired torque whenthe digit contacts the ground.

V. CENTER OF PRESSURE CONTROL

One of the primary capabilities of the emulator system isthe ability to modulate the ground reaction force origin andmagnitude. Each digit of the emulator can sense longitudinalbending forces, which allows for independent modulation offorce produced by each digit. The total ground reaction forceis then defined as the vector sum of the forces produced byeach digit, with center of pressure being the origin of the totalground reaction force. However, the emulator lacks the abilityto sense axial forces, limiting our ability to fully estimate andcontrol the center of pressure and total ground reaction forcemagnitude.

A. Scope of Measurement and Control

This emulator does not have full control over the totalground reaction force vector. Such a device would requireat least five independent modes of actuation in order tocontrol the five components of the total force vector: thecenter of pressure in two dimensions in the ground planeand the direction of the force vector in three dimensions.

These requirements would likely result in a heavier prosthesisdesign. Instead, we have made some simplifying assumptionsand focused on controlling only three components of the totalforce vector: the center of pressure in two dimensions and thenormal ground reaction force magnitude.

In theory, it is possible to control the above-mentioned threecomponents of the total force vector, given that the emulatorhas three independently controlled digits. To do so accuratelywould require measurements of both axial and perpendicularcomponents of the force produced by each digit (Fig. 9).However, axial forces are not measured by the strain gaugesbonded to each digit. The full bridge configuration of thestrain gauges intentionally rejects axial force readings in orderto provide more accurate measurements on the longitudinalbending experienced by the strain gauges. The force vectorsapplied by each digit can be affected by axial force contri-butions, including portions of the normal force and frictionshear forces. This loss of information results in a deviationbetween the estimated total ground reaction force and thetrue total ground reaction force and consequently affects theaccuracy of estimating center of pressure and ground reactionforce magnitude. Thus the emulator is able to control all threedesired components of the total force vector, but suffers frominaccuracies due to insufficient sensing.

There are several methods to mitigate the inaccurate sensingof axial forces. For one, we could install an additional straingauge bridge on each digit to measure the axial forces.However, the cross-sectional area of the digits coupled with therelatively low expected axial forces results in low maximumstrain values. Small strain variations are more difficult to detectand result in poorer signal-to-noise ratios of the strain gauge.Mounting another set of strain gauges on each digit to detectaxial forces would increase the complexity of the device whileproviding limited benefit. For these reasons we decided toforgo measuring axial forces, thus limiting accurate controlof center of pressure and ground reaction force magnitude.

B. Geometric Model Derivation and TestingDesired digit torques are modulated using a high-level

controller based on a geometric model and whose outputs arepassed to a low-level controller for closed-loop control. Themodel of the high-level controller translates the desired centerof pressure and ground reaction force magnitude into thedesired torques for each of the three digits. The model requiresthree inputs: the desired medio-lateral center of pressure, copx,the desired fore-aft center of pressure, copy, and the desirednormal ground reaction force magnitude, FN . In turn, themodel leads to three outputs: the required torques of the leftand right forefoot digits and the heel digit, τl , τr, and τh,respectively. We define an x-y origin for the center of pressurein the prosthesis frame, such that x = 0 is the medio-lateralcenter of the end-effector and y = 0 is the location of theforefoot digit joints (Fig. 10a). Since center of pressure canonly be measured in the ground plane, we project this pointonto the ground plane in a direction normal to the groundplane (Fig. 10b).

The model dictating the desired torques of the three digitscan be defined in terms of the normal components of forces

7

Fig. 9. The strain gauges are only able to detect perpendicular components offorce generated by the digit, shown in blue. Axial force components, shownin yellow, are undetectable yet still contribute to the total ground reactionforce vector, shown in orange. This loss of information results in a deviationbetween the estimated total ground reaction force and the true total groundreaction force.

produced by each digit, Fl ,Fr,Fh, using a weighted averagecalculation. Only the normal components of the forces areused in the model since it is the only component of force thataffects center of pressure.

copx =Fl(−

w2)+Fr(

w2)+Fh(0)

Fl +Fr +Fh(1)

copy =(Fl +Fr)y f +Fhyh

Fl +Fr +Fh(2)

FN = Fl +Fr +Fh (3)

where w is the width between the forefoot digits, y f is theaverage y-coordinate of the two forefoot digits yl and yr, and yhis the y-coordinate of the heel digit (Fig. 10). The y-coordinateof the heel joint is y= dcosθb, where d is the distance betweenthe forefoot joints and the heel joint. The y-coordinate of thedigits are given as:

yl = r f cos(θl −θb) (4)

yr = r f cos(θr −θb) (5)

yh = dcosθb − rhcos(θb +θh) (6)

where r f and rh are lengths of the forefoot digits and heeldigit, respectively. Inverting Eq. 1-3 gives desired digit forcesin terms of constants and inputs:

Fh =FN(copy − y f )

yh − y f(7)

xy

θr

yz

rh cos(θ + θ )b h

r

f

hfb

h

h

h

b

b

y

y

y

w

rθ

y

r

θ

θ

d cos θ

(a) (b)

(c)

yz

Fig. 10. The bottom view of the end-effector is shown along with the threecontact points (a). If the end-effector frame is parallel to the ground (θb = 0),then the origin is located such that x = 0 is the medio-lateral center of theend-effector and y= 0 is the location of the forefoot digit joints. The side viewof the end-effector is shown, with forefoot digit geometry labeled to illustratethe derivation of Eq. 4 and Eq. 5 (b). The side view of the end-effectoris repeated, with heel digit geometry labeled to illustrate the derivation ofEq. 6 (c). Constants include w,d,rt ,rh while the remaining variables can bemeasured or calculated.

Fr =FNcopx

w+

FN −Fh

2(8)

Fl = FN −Fr −Fh (9)

We can now calculate the torques required to generate thedesired normal force for each digit. As previously mentioned,axial forces affect the total force magnitude but are unde-tectable without another set of strain gauges. We tested threesimplifying assumptions about the axial force components ofeach digit, including: assuming axial forces are zero, assumingaxial forces such that the force vector for each digit is normalto the ground, and assuming axial forces such that the forcevector for each digit points along the user’s leg. We foundthe third assumption resulted in the least center of pressureerror across various conditions and used this assumption tocalculate the required torques for each digit:

τl =Flr f cos(θl)

cos(θb)(10)

8

τr =Frr f cos(θr)

cos(θb)(11)

τh =Fhrhcos(θh)

cos(θb)(12)

For the presented emulator design, model constants w,d,r f andrh are equal to 0.074, 0.035, 0.15, 0.089 m, respectively. Theseconstants are taken from the geometry of the end-effector asdescribed in Section II A.

To quantify the error incurred by our geometric model, weprogrammed the emulator to trace out a prescribed centerof pressure pattern. An instrumented treadmill recorded thetrue center of pressure trajectory which we then comparedagainst the desired pattern. We performed the pattern tracingexperiment under three different conditions: one with the end-effector fixed in place over the treadmill, one with a participantwith unilateral transtibial amputation standing while wearingthe end-effector, and one with the same participant walkingwith the end-effector. The instrumented treadmill reports cen-ter of pressure relative to the treadmill origin while the desiredcenter of pressure is defined relative to the end-effector origin.We recorded positions of four reflective markers placed onthe end-effector using an 8-camera motion capture system(frame rate: 100Hz; Vicon Motion Systems, Oxford, UK).This allowed us to compare the desired and actual center ofpressure patterns within the same reference frame as definedby the end-effector. The center of pressure patterns used forall three conditions were defined by first-, second-, and third-order polynomials which closely resembled the natural centerof pressure path seen during walking in healthy adults [41].In the fixed and standing conditions, the polynomial patternswere traced in both directions in 1 s (Fig. 12). In the walkingcondition, the polynomial patterns were only traced in theforward direction and were timed according to the averagestance duration of the participant. In the fixed and standingconditions, we additionally traced a block-letter ‘S’ in orderto further quantify the accuracy of center of pressure control.In both of these conditions, the block-letter ‘S’ was traced in10 s.

C. Results

The maximum length and width of the support triangle is0.20 m and 0.074 m and the peak total vertical force magnitudegenerated by the emulator is 2.3 kN (Fig. 11). These resultsare achieved when the end-effector is as low to the groundas possible, with each digit applying peak torque. In config-urations where the end-effector is higher off the ground, θ fand θh increase, resulting in a smaller support triangle anda smaller vertical component of the ground reaction forceof each digit. For a given support triangle, vertical forcemagnitude decreases as the center of pressure approaches thecorners of the triangle (Fig. 11). When the center of pressure isin the middle of the support triangle, the total ground reactionforce receives contributions from all three digits. However,when the center of pressure is at one vertex of the support

2327 N

632 N

0.20 m

0.074 m

Fig. 11. The area between the ground contact pads designates the supporttriangle of the end-effector. The contour plot depicts the peak controllableforce magnitude, with lighter colors indicating higher forces.

triangle, the total ground reaction force can only receive acontribution from a single digit, which limits the peak groundreaction force magnitude.

There was a strong correlation between desired and mea-sured center of pressure trajectories (Fig. 12), indicating thatthe geometric model provided an accurate mapping from de-sired center of pressure to required digit torques. We calculatedRMS error values as the Euclidean distance between theaverage measured center of pressure location and the desiredcenter of pressure location at each instance in time. With theend-effector fixed in place over the treadmill, the RMS errorwas 14 mm across all six polynomial patterns. The RMS errorincreased to an average of 15 and 18 mm for standing andwalking conditions, respectively. The RMS error was 8 mmfor the block-letter ‘S’ in both fixed and standing conditions.These errors could be due to several factors, discussed in moredetail in the Discussion section.

The center of pressure control described only applies whenall three digits are in contact with the ground. To test thecapability of the emulator to maintain full ground contact,we modified a spring controller to maximize ground con-tact during stance. Upon detecting heel strike, the controllercommands a desired torque to the forefoot digits in orderto obtain full ground contact. As stance progresses, the heeldigit attempts to maintain a minimum of 20 Nm. During latestance the heel digit is too short to reach the ground and theemulator enters a period where only the two forefoot digits arein contact with the ground. During a walking trial with 133steps, the emulator was able to achieve full ground contactfor 56%±2.7% of stance. The heel digit had ground contactfor the first 70%±3.1% of stance and the forefoot digits hadground contact for the last 86%±1.4% of stance.

During the initial portion of stance when only the heel digitis in contact with the ground, the center of pressure is limitedto the contact patch of the heel digit. During the final portionof stance when only the two forefoot digits are in contact withthe ground, the center of pressure lies on the line between thecontact points of the two digits. The center of pressure can beroughly controlled along the line, but controllability is reducedas the angle of the digit relative to the ground increases.

VI. DISCUSSION

We designed, built, and tested a prosthesis emulator in-tended to assist in the development of controllers to enhanceuser balance and improve energy expenditure. Benchtop test-ing demonstrated high peak torques and bandwidths necessary

9

RMSE=15mmt=1.0s

RMSE=14mmt=1.0s

RMSE=19mmt=0.73s

RMSE=14mmt=1.0s

RMSE=14mmt=1.0s

RMSE=17mmt=0.75s

RMSE=16mmt=1.0s

RMSE=14mmt=1.0s

RMSE=19mmt=0.76s

RMSE=14mmt=1.0s

RMSE=14mmt=1.0s

RMSE=16mmt=0.76s

RMSE=17mmt=1.0s

RMSE=14mmt=1.0s

RMSE=18mmt=0.76s

RMSE=16mmt=1.0s

RMSE=14mmt=1.0s

RMSE=16mmt=0.77s

RMSE=8mmt=10s

RMSE=8mmt=10s

Fixe

dSt

andi

ngW

alki

ng Desired

Measured

200 mm

200

mm

60 mm

75 m

m

Fig. 12. Desired and measured center of pressure patterns. The top, middle, and bottom rows indicate patterns traced with the end-effector during walking,standing, and fixed conditions, respectively. The arrows indicate direction(s) of center of pressure movement with RMS error values and time taken to draweach pattern listed above each plot. RMS error values were calculated as the Euclidean distance between the average measured center of pressure locationand the desired center of pressure location at each instance in time. The shaded grey triangles indicate the support triangle of the end-effector relative to thedesired pattern. Instances where the measured center of pressure lies outside of the shaded triangle are due to misalignment errors, addressed in the Discussionsection.

for high fidelity behavior emulation. In addition, the emulatoris capable of modulating the location and magnitude of thetotal ground reaction force during stance. Testing duringtreadmill walking revealed accurate torque tracking and centerof pressure control during standing and walking.

One advantage of the prosthesis emulator, compared toprevious powered prostheses, is its ability to modulate cen-ter of pressure location and normal ground reaction forcemagnitude. Force modulation at each digit can be achievedwith a simple lever based mechanism based on our previousdesigns [30], [28]. These mechanisms result in lightweightend-effectors which allow for rapid iteration and testing ofvarious control approaches without the need to redesign andrebuild new hardware. Furthermore, tethering the end-effectorallows the worn mass to be similar to lightweight semi-activedevices (e.g. [42]), other tethered devices (e.g. [43]), andeven lighter than untethered active devices (e.g. [39], [44]).Existing designs that provide actuation in the sagittal andfrontal planes are based on a carbon keel hinged on a universaljoint [43], [45]. Another design uses a four-bar linkage to shiftthe center of pressure in the sagittal plane [46]. Devices basedon carbon keels preserve the rollover shape and user familiarityof commercial feet, but often incur complex mechanisms andloading scenarios, potentially resulting in larger structures withhigher mass. In contrast, the design of the proposed prosthesisemulator results in a worn mass similar to the weight of anaverage male foot, while exceeding the range of motion andpower requirements of normal walking. Additionally, having

three discrete contact points also allows for the potential ofcenter of pressure control over uneven terrain.

Desired and measured center of pressure trajectories showedRMS error values of, at most, 19 mm, or 10% of full trajectorylength. There are several sources of error outside of thegeometric model that contribute to this error in trajectories.These include misalignment when defining the origin of themotion capture system, offsets in center of pressure due tothe width of the ground contact pad on each digit, and errorswhen tracking desired torque for each digit. For one, there areinherent discrepancies between the center of pressure recordedby the instrumented treadmill and the motion capture systems.To quantify these errors, we outfitted a 1 m pole with reflectivemarkers and applied point loads to the treadmill, similar to[47]. We discovered center of pressure errors ranging from0.70 to 8.1 mm across the 12 locations we sampled on thetreadmill. Center of pressure tracking error may also be dueto the width of the ground contact pads on each digit, whichmeasure 2.5 cm in width. Thus, small changes in the roll angleof the end-effector could result in a shifting center of pressurelocation for each digit individually, which affects the centerof pressure location for the end-effector as a whole. Center ofpressure tracking error can also arise from inaccurate trackingof the digit torques dictated by the geometric model, whichis largely due to phase lag from the low-level controller orexternal perturbations. When observing the patterns traced inthe fixed condition, the center of pressure RMS error valueswere much lower for the ‘S’ pattern (0.25 Nm, 10 s trace

10

time) compared to the polynomial patterns (average 1.0 Nm,1.0 s trace time). This is largely due to the fact that desiredtorque is changing quickly in the polynomial patterns, resultingin a phase lag from the low-level controller (Fig. 7c). Ifphase lag is ignored, RMS error is likely to be even smaller,since the measured center of pressure and desired trajectoriesare geometrically highly comparable. External perturbations,such as variability introduced during human walking, furthercontribute to torque tracking error. For example, walking trialswith our participant resulted in torque tracking RMS errorof 3.6 Nm, even though the average trace time was slowerat 0.76 s for a unidirectional pattern compared to 1.0 s for abidirectional pattern.

Limited instrumentation on the device is one limitation ofthe described prosthesis emulator. In particular, the geometricmodel needs to make simplifying assumptions about the axialforce components on each digit, since these are undetectablewith the strain gauge configuration used. As a result, wecannot perfectly control center of pressure location and groundreaction force magnitude. Together with other sources of errorpreviously discussed,limited instrumentation leads to RMScenter of pressure tracking errors that range from 8 mm to19 mm across a variety of conditions and center of pressurepatterns. Assumptions made by the geometric model mayresult in larger errors during conditions where axial forcecomponents are higher, or in configurations where the end-effector is placed at large roll or pitch angles.

Future prosthesis emulator iterations may want to considerseveral design changes to improve device performance. Forone, error currently present in center of pressure control accu-racy could be mitigated by installing axial strain gauges. To dothis, the digits would need to be redesigned to reduce the axialstrength of each digit without compromising on longitudinalbending strength. Such constraints would likely dictate a morecomplex design of the digits and could prove less robust. Digitrange of motion could also be improved, particularly duringplantarflexion. A larger digit range of motion could improvecenter of pressure control, as digits would be in contact withthe ground for a larger percentage of stance. In the currentdevice design, certain behaviors may cause the forefoot digitsto extend to the point of coming in contact with the hard stops.This could be addressed by redesigning the end-effector frameto allow for higher plantarflexion angles, but would result inincreased device weight and size.

VII. CONCLUSION

This ankle-foot prosthesis emulator enables rapid develop-ment and testing of controllers intended to advance our under-standing of the biomechanics of individuals with amputation.We are currently planning on using this emulator to investigatethe effects of balance on energy expenditure and to identifybeneficial control behaviors. These insights can potentiallyaid in the development of future prosthetic devices, increasemobility and improve the quality of life of individuals withamputation.

ACKNOWLEDGMENT

The authors thank Josh Caputo, Myunghee Kim, GeneMerewether, and members of the Stanford BiomechatronicsLaboratory for helpful discussions and advice on the designof the prosthesis emulator, as well as Susan Stenman, CP, forassistance with fitting of the device to our participant.

CONFLICT OF INTEREST

The authors declare no potential conflicts of interest withrespect to the research, authorship, and/or publication of thisarticle.

REFERENCES

[1] K. Ziegler-Graham, E. J. MacKenzie, P. L. Ephraim, T. G. Travison, andR. Brookmeyer, “Estimating the prevalence of limb loss in the unitedstates: 2005 to 2050,” Archives of physical medicine and rehabilitation,vol. 89, no. 3, pp. 422–429, 2008.

[2] L. Hak, J. H. van Dieen, P. van der Wurff, and H. Houdijk,“Stepping Asymmetry Among Individuals With Unilateral TranstibialLimb Loss Might Be Functional in Terms of Gait Stability,” PhysicalTherapy, vol. 94, no. 10, pp. 1480–1488, 2014. [Online]. Available:https://academic.oup.com/ptj/article-lookup/doi/10.2522/ptj.20130431

[3] H. Houdijk, E. Pollmann, M. Groenwald, H. Wiggerts, W. Polomski,M. Groenewold, H. Wiggerts, W. Polomski, M. Groenwald, H. Wiggerts,and W. Polomski, “The energy cost for the step-to-step transition inamputee walking,” Gait and Posture, vol. 30, no. 1, pp. 35–40, 2009.

[4] J. Paysant, C. Beyaert, A.-M. Dati, N. Martinet, and J.-M. Andr,“Influence of terrain on metabolic and temporal gait characteristics ofunilateral transtibial amputees,” The Journal of Rehabilitation Researchand Development, vol. 43, no. 2, p. 153, 2006. [Online]. Available:http://www.rehab.research.va.gov/jour/06/43/2/pdf/paysant.pdf

[5] D. H. Gates, J. B. Dingwell, S. J. Scott, E. H. Sinitski, andJ. M. Wilken, “Gait characteristics of individuals with transtibialamputations walking on a destabilizing rock surface,” Gait andPosture, vol. 36, no. 1, pp. 33–39, 2012. [Online]. Available:http://dx.doi.org/10.1016/j.gaitpost.2011.12.019

[6] A. L. Hof, R. M. van Bockel, T. Schoppen, and K. Postema, “Controlof lateral balance in walking: experimental findings in normal subjectsand above-knee amputees,” Gait & posture, vol. 25, no. 2, pp. 250–258,2007.

[7] R. Beurskens, J. M. Wilken, and J. B. Dingwell, “Dynamicstability of superior vs. inferior body segments in individuals withtranstibial amputation walking in destabilizing environments,” Journalof Biomechanics, vol. 47, no. 12, pp. 3072–3079, 2014. [Online].Available: http://dx.doi.org/10.1016/j.jbiomech.2014.06.041

[8] D. H. Gates, S. J. Scott, J. M. Wilken, and J. B. Dingwell, “Frontalplane dynamic margins of stability in individuals with and withouttranstibial amputation walking on a loose rock surface,” Gait andPosture, vol. 38, no. 4, pp. 570–575, 2013. [Online]. Available:http://dx.doi.org/10.1016/j.gaitpost.2013.01.024

[9] W. C. Miller, A. Deathe, M. Speechley, and J. Koval, “The influenceof falling, fear of falling, and balance confidence on prostheticmobility and social activity among individuals with a lower extremityamputation,” Archives of Physical Medicine and Rehabilitation,vol. 82, no. 9, pp. 1238–1244, sep 2001. [Online]. Available:http://linkinghub.elsevier.com/retrieve/pii/S0003999301150264

[10] S. M. Felcher, D. J. Stinner, C. A. Krueger, J. M. Wilken,D. A. Gajewski, and J. R. Hsu, “Falls in a Young ActiveAmputee Population: A Frequent Cause of Rehospitalization?” MilitaryMedicine, vol. 180, no. 10, pp. 1083–1086, 2015. [Online]. Available:http://publications.amsus.org/doi/10.7205/MILMED-D-14-00450

[11] J. M. Donelan, R. Kram, and A. D. Kuo, “Mechanical work for step-to-step transitions is a major determinant of the metabolic cost of humanwalking,” Journal of Experimental Biology, vol. 205, no. 23, pp. 3717–3727, 2002.

[12] S. K. Au, P. Dilworth, and H. Herr, “An ankle-foot emulation system forthe study of human walking biomechanics,” in Robotics and Automation,2006. ICRA 2006. Proceedings 2006 IEEE International Conference on.IEEE, 2006, pp. 2939–2945.

[13] J. J. Rice, J. M. Schimmels, and S. Huang, “Design and evaluation of apassive ankle prosthesis with powered push-off,” Journal of Mechanismsand Robotics, vol. 8, no. 2, p. 021012, 2016.

11

[14] B. Brackx, M. Van Damme, A. Matthys, B. Vanderborght, andD. Lefeber, “Passive ankle-foot prosthesis prototype with extended push-off,” International Journal of Advanced Robotic Systems, vol. 10, 2013.

[15] H. M. Herr and A. M. Grabowski, “Bionic ankle–foot prosthesisnormalizes walking gait for persons with leg amputation,” Proc. R. Soc.B, vol. 279, no. 1728, pp. 457–464, 2012.

[16] R. E. Quesada, J. M. Caputo, and S. H. Collins, “Increasing anklepush-off work with a powered prosthesis does not necessarily reducemetabolic rate for transtibial amputees,” Journal of Biomechanics,vol. 49, no. 14, pp. 3452–3459, 2016. [Online]. Available:http://dx.doi.org/10.1016/j.jbiomech.2016.09.015

[17] P. Malcolm, R. E. Quesada, J. M. Caputo, and S. H. Collins, “Theinfluence of push-off timing in a robotic ankle-foot prostheses on theenergetics and mechanics of walking,” Journal of NeuroEngineering andRehabilitation, vol. 12, no. 21, pp. 1–14, 2015.

[18] M. Kim and S. H. Collins, “Once-per-step control of ankle-foot prosthe-sis push-off work reduces effort associated with balance during walking,”Journal of NeuroEngineering and Rehabilitation, vol. 12, no. 1, p. 43,2015.

[19] T. Ijmker, S. Noten, C. J. Lamoth, P. J. Beek, L. H. V. van derWoude, and H. Houdijk, “Can external lateral stabilization reduce theenergy cost of walking in persons with a lower limb amputation?” Gaitand Posture, vol. 40, no. 4, pp. 616–621, 2014. [Online]. Available:http://dx.doi.org/10.1016/j.gaitpost.2014.07.013

[20] M. Kim and S. H. Collins, “Step-to-Step Ankle Inver-sion/Eversion Torque Modulation Can Reduce Effort Asso-ciated with Balance,” Frontiers in Neurorobotics, vol. 11,no. November, pp. 1–11, 2017. [Online]. Available:http://journal.frontiersin.org/article/10.3389/fnbot.2017.00062/full

[21] D. P. Ferris, K. E. Gordon, G. S. Sawicki, and A. Peethambaran, “Animproved powered ankle–foot orthosis using proportional myoelectriccontrol,” Gait & posture, vol. 23, no. 4, pp. 425–428, 2006.

[22] J. F. Veneman, R. Kruidhof, E. E. Hekman, R. Ekkelenkamp, E. H.Van Asseldonk, and H. Van Der Kooij, “Design and evaluation ofthe lopes exoskeleton robot for interactive gait rehabilitation,” IEEETransactions on Neural Systems and Rehabilitation Engineering, vol. 15,no. 3, pp. 379–386, 2007.

[23] C. Abul-Haj and N. Hogan, “An emulator system for developingimproved elbow-prosthesis designs,” IEEE transactions on biomedicalengineering, no. 9, pp. 724–737, 1987.

[24] W. C. Flowers and R. W. Mann, “An electrohydraulic knee-torque con-troller for a prosthesis simulator,” Journal of biomechanical engineering,vol. 99, no. 1, pp. 3–8, 1977.

[25] J. M. Caputo and S. H. Collins, “An experimental robotic testbedfor accelerated development of ankle prostheses,” in Robotics andautomation (ICRA), 2013 IEEE international conference on. IEEE,2013, pp. 2645–2650.

[26] K. A. Witte, J. Zhang, R. W. Jackson, and S. H. Collins, “Design of twolightweight, high-bandwidth torque-controlled ankle exoskeletons,” inRobotics and Automation (ICRA), 2015 IEEE International Conferenceon. IEEE, 2015, pp. 1223–1228.

[27] J. Zhang, P. Fiers, K. A. Witte, R. W. Jackson, K. L. Poggensee, C. G.Atkeson, and S. H. Collins, “During Walking,” Science, vol. 1284, no.June, pp. 1280–1284, 2017.

[28] S. H. Collins, M. Kim, T. Chen, and T. Chen, “An ankle-foot prosthesisemulator with control of plantarflexion and inversion-eversion torque,” inRobotics and Automation (ICRA), 2015 IEEE International Conferenceon. IEEE, 2015, pp. 1210–1216.

[29] R. W. Jackson and S. H. Collins, “An experimentalcomparison of the relative benefits of work and torqueassistance in ankle exoskeletons,” Journal of Applied Physiology,vol. 119, no. 5, pp. 541–557, 2015. [Online]. Available:http://jap.physiology.org/lookup/doi/10.1152/japplphysiol.01133.2014

[30] J. M. Caputo and S. H. Collins, “A universal ankle–foot prosthesisemulator for human locomotion experiments,” Journal of biomechanicalengineering, vol. 136, no. 3, p. 035002, 2014.

[31] M. R. Hawes and D. Sovak, “Quantitative morphology of the humanfoot in a north american population,” Ergonomics, vol. 37, no. 7, pp.1213–1226, 1994.

[32] D. A. Winter, Biomechanics and Motor Control of Human Movement,2nd ed. Toronto, Canada: John Wiley & Sons, Inc., 1990.

[33] A. H. Shultz, J. E. Mitchell, D. Truex, B. E. Lawson, and M. Goldfarb,“Preliminary evaluation of a walking controller for a powered ankleprosthesis,” in Robotics and automation (ICRA), 2013 IEEE interna-tional conference on. IEEE, 2013, pp. 4838–4843.

[34] M. F. Eilenberg, H. Geyer, and H. Herr, “Control of a powered ankle–foot prosthesis based on a neuromuscular model,” IEEE transactions

on neural systems and rehabilitation engineering, vol. 18, no. 2, pp.164–173, 2010.

[35] M. K. Shepherd and E. J. Rouse, “The VSPA Foot: A Quasi-PassiveAnkle-Foot Prosthesis with Continuously Variable Stiffness,” IEEETransactions on Neural Systems and Rehabilitation Engineering, vol. 25,no. 12, pp. 2375–2386, 2017.

[36] S. H. Collins and A. D. Kuo, “Recycling energy to restore impairedankle function during human walking,” PLoS ONE, vol. 5, no. 2, p.e9307, 2010.

[37] J. Zhang, C. C. Cheah, and S. H. Collins, “Torque control in leggedlocomotion,” in Bioinspired Legged Locomotion: Models, Concepts,Control and Applications, 1st ed., M. A. Sharbafi and A. Seyfarth, Eds.Butterworth-Heinemann, 2017, ch. 5, pp. 347–395.

[38] R. Versluys, G. Lenaerts, M. Van Damme, I. Jonkers, A. Desomer,B. Vanderborght, L. Peeraer, G. Van Der Perre, and D. Lefeber, “Suc-cessful preliminary walking experiments on a transtibial amputee fittedwith a powered prosthesis,” Prosthetics and Orthotics International,vol. 33, no. 4, pp. 368–377, 2009.

[39] S. Au, M. Berniker, and H. Herr, “Powered ankle-foot prosthesis toassist level-ground and stair-descent gaits,” Neural Networks, vol. 21,no. 4, pp. 654–666, 2008.

[40] P. G. Griffiths, R. B. Gillespie, and J. S. Freudenberg, “A fundamentallinear systems conflict between performance and passivity in hapticrendering,” IEEE Transactions on robotics, vol. 27, no. 1, pp. 75–88,2011.

[41] D. A. Winter, “Human balance and posture control during standing andwalking,” Gait & posture, vol. 3, no. 4, pp. 193–214, 1995.

[42] M. Cempini, L. J. Hargrove, and T. Lenzi, “Design, Development,and Bench-top Testing of a Powered Polycentric Ankle Prosthesis,”IEEE/RSJ International Conference on Intelligent Robots and System,no. September, pp. 1–6, 2017.

[43] E. M. Ficanha, G. A. Ribeiro, H. Dallali, and M. Rastgaar,“Design and Preliminary Evaluation of a Two DOFsCable-Driven Ankle-Foot Prosthesis with Active Dorsiflexion-Plantarflexion and Inversion-Eversion.” Frontiers in bioengineeringand biotechnology, vol. 4, no. May, p. 36, 2016.[Online]. Available: http://www.ncbi.nlm.nih.gov/pubmed/27200342http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=PMC4852183

[44] S. Huang, J. P. Wensman, and D. P. Ferris, “An experimental poweredlower limb prosthesis using proportional myoelectric control,” Journalof Medical Devices, vol. 8, no. 2, p. 024501, 2014.

[45] R. D. Bellman, M. A. Holgate, and T. G. Sugar, “Sparky 3: Design of anactive robotic ankle prosthesis with two actuated degrees of freedom us-ing regenerative kinetics,” in Biomedical Robotics and Biomechatronics,2008. BioRob 2008. 2nd IEEE RAS & EMBS International Conferenceon. IEEE, 2008, pp. 511–516.

[46] A. K. LaPre, R. D. Wedge, B. R. Umberger, and F. C. Sup, “Preliminarystudy of a robotic foot-ankle prosthesis with active alignment,” inRehabilitation Robotics (ICORR), 2017 International Conference on.IEEE, 2017, pp. 1299–1304.

[47] S. H. Collins, P. G. Adamczyk, D. P. Ferris, and A. D. Kuo, “Asimple method for calibrating force plates and force treadmills usingan instrumented pole,” Gait & Posture, vol. 29, no. 1, pp. 59–64, 2009.